Arc presentations of knots and links Hwa Jeong Lee (KAIST) December 10, 2013 IBS Table of Contents 1 What is a Knot? 2 Arc presentation and Arc index 3 Arc index of Kanenobu knots (joint with Hideo Takioka) 4 A relation between arc index and Thurston-Bennequin number Table of Contents 1 What is a Knot? 2 Arc presentation and Arc index 3 Arc index of Kanenobu knots (joint with Hideo Takioka) 4 A relation between arc index and Thurston-Bennequin number Table of Contents 1 What is a Knot? 2 Arc presentation and Arc index 3 Arc index of Kanenobu knots (joint with Hideo Takioka) 4 A relation between arc index and Thurston-Bennequin number Table of Contents 1 What is a Knot? 2 Arc presentation and Arc index 3 Arc index of Kanenobu knots (joint with Hideo Takioka) 4 A relation between arc index and Thurston-Bennequin number What is Knot Theory? Unknottedcircle Knottedcircle Unknottedcircle Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory? Unknottedcircle Knottedcircle Unknotted circle Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory? Unknottedcircle Knottedcircle Unknottedcircle Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory? Unknottedcircle Knottedcircle Unknotted circle Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory? Unknottedcircle Knottedcircle Unknottedcircle Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory? Unknottedcircle Knottedcircle Unknottedcircle Knot theory in the mathematical sense is the study of closed curves in R3. What is a Knot? • A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3. Main Problem How to classify knots? What is a Knot? • A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3. Unknot Main Problem How to classify knots? What is a Knot? • A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3. Unknot Trefoil knot Main Problem How to classify knots? What is a Knot? • A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3. Unknot Trefoil knot Main Problem How to classify knots? What is a Knot? • A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3. Unknot Trefoil knot Hopf link Main Problem How to classify knots? What is a Knot? • A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3. Unknot Trefoil knot Hopf link Main Problem How to classify knots? Two knots are equivalent 3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2. Knots that are equivalent can be regarded as being simply the same (type) knot. Question 1. How do we know these are actually same or different knots ? Two knots are equivalent 3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2. Knots that are equivalent can be regarded as being simply the same (type) knot. Question 1. How do we know these are actually same or different knots ? Two knots are equivalent 3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2. Knots that are equivalent can be regarded as being simply the same (type) knot. Question 1. ∼ , ≁ How do we know these are actually same or different knots ? Two knots are equivalent 3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2. Knots that are equivalent can be regarded as being simply the same (type) knot. Question 1. ∼ , ≁ How do we know these are actually same or different knots ? Knot diagram • A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass • A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Knot diagram • A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass • A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Knot diagram • A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass • A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Knot diagram • A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass • A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Knot diagram • A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass • A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Reidemeister Theorem Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies. Reidemeister Theorem Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies. I I Reidemeister Theorem Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies. I I II II Reidemeister Theorem Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies. I I II II III Reidemeister Theorem Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies. I I II II III Example II II I Are two diagrams equivalent? ∼? Are two diagrams equivalent? ∼? ∼? Knot Invariant • A knot invariant is a quantity defined for each knot which is the same for equivalent knots. • A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant • A knot invariant is a quantity defined for each knot which is the same for equivalent knots. • A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant • A knot invariant is a quantity defined for each knot which is the same for equivalent knots. • A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant • A knot invariant is a quantity defined for each knot which is the same for equivalent knots. • A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant • A knot invariant is a quantity defined for each knot which is the same for equivalent knots.